The number of abundant elements in union-closed families without small sets

Abstract

We let $\mathcal{F}$ be a finite family of sets closed under taking unions and $\emptyset \not \in \mathcal{F}$, and call an element abundant if it belongs to more than half of the sets of $\mathcal{F}$. In this notation, the classical Frankl's conjecture (1979) asserts that $\mathcal{F}$ has an abundant element. As possible strengthenings, Poonen (1992) conjectured that if $\mathcal{F}$ has precisely one abundant element, then this element belongs to each set of $\mathcal{F}$, and Cui and Hu (2019) investigated whether $\mathcal{F}$ has at least $k$ abundant elements if a smallest set of $\mathcal{F}$ is of size at least $k$. Cui and Hu conjectured that this holds for $k = 2$ and asked whether this also holds for the cases $k = 3$ and $k > \frac{n}{2}$ where $n$ is the size of the largest set of $\mathcal{F}$. We show that $\mathcal{F}$ has at least $k$ abundant elements if $k \geq n - 3$, and that $\mathcal{F}$ has at least $k - 1$ abundant elements if $k = n - 4$, and we construct a union-closed family with precisely $k - 1$ abundant elements for every $k$ and $n$ satisfying $n - 4 \geq k \geq 3$ and $n \geq 9$ (and for $k = 3$ and $n = 8$). We also note that $\mathcal{F}$ always has at least $\min \{ n, 2k - n + 1 \}$ abundant elements. On the other hand, we construct a union-closed family with precisely two abundant elements for every $k$ and $n$ satisfying $n \geq \max \{ 3, 5k-4 \}$. Lastly, we show that Cui and Hu's conjecture for $k = 2$ stands between Frankl's conjecture and Poonen's conjecture.